feat(library/data/set/function.lean): begin theory of functions on sets

library/data/set/basic.lean

@@ -1,6 +1,6 @@
 /-
-copyright (c) 2014 Jeremy Avigad. All rights reserved.
-Released under Apache 2.0 license as described in the file LIcENSE.
+Copyright (c) 2014 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
 
 Module: data.set.basic
 Author: Jeremy Avigad, Leonardo de Moura
@@ -22,16 +22,16 @@ notation e ∈ a := mem e a
 theorem setext {a b : set T} (H : ∀x, x ∈ a ↔ x ∈ b) : a = b :=
 funext (take x, propext (H x))
 
-definition subset (a b : set T) := ∀ x, x ∈ a → x ∈ b
+definition subset (a b : set T) := ∀⦃x⦄, x ∈ a → x ∈ b
 infix `⊆`:50 := subset
 
 /- bounded quantification -/
 
-abbreviation bounded_forall (a : set T) (P : T → Prop) := ∀x, x ∈ a → P x
+abbreviation bounded_forall (a : set T) (P : T → Prop) := ∀⦃x⦄, x ∈ a → P x
 notation `forallb` binders `∈` a `,` r:(scoped:1 P, P) := bounded_forall a r
 notation `∀₀` binders `∈` a `,` r:(scoped:1 P, P) := bounded_forall a r
 
-abbreviation bounded_exists (a : set T) (P : T → Prop) := ∃x, x ∈ a ∧ P x
+abbreviation bounded_exists (a : set T) (P : T → Prop) := ∃⦃x⦄, x ∈ a ∧ P x
 notation `existsb` binders `∈` a `,` r:(scoped:1 P, P) := bounded_exists a r
 notation `∃₀` binders `∈` a `,` r:(scoped:1 P, P) := bounded_exists a r
 

library/data/set/function.lean

@@ -0,0 +1,64 @@
+/-
+Copyright (c) 2014 Jeremy Avigad. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+
+Module: data.set.function
+Author: Jeremy Avigad, Andrew Zipperer
+
+Functions between subsets of finite types.
+-/
+import .basic
+import algebra.function
+open function eq.ops
+
+namespace set
+  variables {X Y Z : Type}
+
+  abbreviation eq_on (f1 f2 : X → Y) (a : set X) : Prop :=
+  ∀₀ x ∈ a, f1 x = f2 x
+
+  definition image (f : X → Y) (a : set X) : set Y := {y : Y | ∃x, x ∈ a ∧ f x = y}
+  notation f `'[`:max a `]` := image f a
+
+  theorem image_eq_image_of_eq_on {f1 f2 : X → Y} {a : set X} (H1 : eq_on f1 f2 a) :
+    f1 '[a] = f2 '[a] :=
+  setext (take y, iff.intro
+    (assume H2,
+      obtain x (H3 : x ∈ a ∧ f1 x = y), from H2,
+      have H4 : x ∈ a, from and.left H3,
+      have H5 : f2 x = y, from (H1 H4)⁻¹ ⬝ and.right H3,
+      exists.intro x (and.intro H4 H5))
+    (assume H2,
+      obtain x (H3 : x ∈ a ∧ f2 x = y), from H2,
+      have H4 : x ∈ a, from and.left H3,
+      have H5 : f1 x = y, from (H1 H4) ⬝ and.right H3,
+      exists.intro x (and.intro H4 H5)))
+
+  definition maps_to (f : X → Y) (a : set X) (b : set Y) : Prop := ∀⦃x⦄, x ∈ a → f x ∈ b
+
+  theorem maps_to_compose {g : Y → Z} {f : X → Y} {a : set X} {b : set Y} {c : set Z}
+     (H1 : maps_to g b c) (H2 : maps_to f a b) : maps_to (g ∘ f) a c :=
+  take x, assume H : x ∈ a, H1 (H2 H)
+
+  definition inj_on (f : X → Y) (a : set X) : Prop :=
+  ∀⦃x1 x2 : X⦄, x1 ∈ a → x2 ∈ a → f x1 = f x2 → x1 = x2
+
+  theorem inj_on_of_eq_on {f1 f2 : X → Y} {a : set X} (inj_f1 : inj_on f1 a)
+      (eq_f1_f2 : eq_on f1 f2 a) : inj_on f2 a :=
+  take x1 x2 : X,
+  assume ax1 : x1 ∈ a,
+  assume ax2 : x2 ∈ a,
+  assume H : f2 x1 = f2 x2,
+  have H' : f1 x1 = f1 x2, from eq_f1_f2 ax1 ⬝ H ⬝ (eq_f1_f2 ax2)⁻¹,
+  show x1 = x2, from inj_f1 ax1 ax2 H'
+
+  definition surj_on (f : X → Y) (a : set X) (b : set Y) : Prop := b ⊆ f '[a]
+
+  theorem surj_on_of_eq_on {f1 f2 : X → Y} {a : set X} {b : set Y} (surj_f1 : surj_on f1 a b)
+      (eq_f1_f2 : eq_on f1 f2 a) : surj_on f2 a b :=
+  take y, assume H : y ∈ b,
+  obtain x (H1 : x ∈ a ∧ f1 x = y), from surj_f1 H,
+  have H2 : x ∈ a, from and.left H1,
+  have H3 : f2 x = y, from (eq_f1_f2 H2)⁻¹ ⬝ and.right H1,
+  exists.intro x (and.intro H2 H3)
+end set